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Parikh vectors
8BXJ_1 1GKM_1 5QPR_1 Letter Amino acid
4 13 6 H Histidine
11 24 14 I Isoleucine
7 18 25 K Lycine
1 27 15 R Arginine
7 26 24 D Aspartic acid
6 18 21 T Threonine
1 3 6 W Tryptophan
10 80 36 A Alanine
8 41 18 G Glycine
2 21 13 P Proline
7 31 35 V Valine
4 71 30 L Leucine
6 10 15 M Methionine
4 11 18 F Phenylalanine
9 39 14 S Serine
7 15 8 N Asparagine
0 6 8 C Cysteine
5 18 16 Q Glutamine
1 29 25 E Glutamic acid
0 8 17 Y Tyrosine 

8BXJ_1|Chain A|Copper ABC transporter substrate-binding protein|Salmonella enterica subsp. enterica serovar Typhimurium (90371)
>1GKM_1|Chain A[auth B]|Histidine ammonia-lyase|Pseudomonas putida (303)
>5QPR_1|Chain A|Farnesyl diphosphate synthase|Trypanosoma cruzi (5693)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
8BXJ , Knot 52 100 0.79 36 81 96 
GAMGMLKHISHGDMNAASDASVQQVIKGTGIVKDIDMNSKKITISHEAIPAVGWPAMTMRFTFVNADDAINALKTGNHVDFSFIQQGNISLLKSINVTQS
1GKM , Knot 200 509 0.81 40 214 459 
TELTLKPGTLTLAQLRAIHAAPVRLQLDASAAPAIDASVACVEQIIAEDRTAYGINTGFGLLASTRIASHDLENLQRSLVLSHAAGIGAPLDDDLVRLIMVLKINSLSRGFSGIRRKVIDALIALVNAEVYPHIPLKGSVGASGDLAPLAHMSLVLLGEGKARYKGQWLSATEALAVAGLEPLTLAAKEGLALLNGTQASTAYALRGLFYAEDLYAAAIACGGLSVEAVLGSRSPFDARIHEARGQRGQIDTAACFRDLLGDSSEVSLSHKNADKVQDPYSLRCQPQVMGACLTQLRQAAEVLGIEANAVSDNPLVFAAEGDVISGGNFHAEPVAMAADNLALAIAEIGSLSERRISLMMDKHMSQLPPFLVENGGVNSGFMIAQVTAAALASENKALSHPHSVDSLPTSANQEDHVSMAPAAGKRLWEMAENTRGVLAIEWLGACQGLDLRKGLKTSAKLEKARQALRSEVAHYDRDRFFAPDIEKAVELLAKGSLTGLLPAGVLPSL
5QPR , Knot 157 364 0.84 40 211 347 
GPMASMERFLSVYDEVQAFLLDQLQSKYEIDPNRARYLRIMMDTTCLGGKYFRGMTVVNVAEGFLAVTQHDEATKERILHDACVGGWMIEFLQAHYLVEDDIMDGSVMRRGKPCWYRFPGVTTQCAINDGIILKSWTQIMAWHYFADRPFLKDLLCLFQKVDYATAVGQMYDVTSMCDSNKLDPEVAQPMTTDFAEFTPAIYKRIVKYKTTFYTYLLPLVMGLLVSEAAASVEMNLVERVAHLIGEYFQVQDDVMDCFTPPEQLGKVGTDIEDAKCSWLAVTFLGKANAAQVAEFKANYGEKDPAKVAVVKRLYSKANLQADFAAYEAEVVREVESLIEQLKVKSPTFAESVAVVWEKTHKRKK

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(8BXJ_1)}(2) \setminus P_{f(1GKM_1)}(2)|=32\), \(|P_{f(1GKM_1)}(2) \setminus P_{f(8BXJ_1)}(2)|=165\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:1111110010010101100101001101011100101000010100011111111110101011010011011001001010110010101100101000
Pair \(Z_2\) Length of longest common subsequence
8BXJ_1,1GKM_1 197 4
8BXJ_1,5QPR_1 202 3
1GKM_1,5QPR_1 161 3

Newick tree

 
[
	8BXJ_1:10.39,
	[
		1GKM_1:80.5,5QPR_1:80.5
	]:24.89
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{609 }{\log_{20} 609}-\frac{100}{\log_{20}100})=149.\)
Status Protein1 Protein2 d d1/2
Query variables 8BXJ_1 1GKM_1 188 111
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]